Translation
Table of Content
2D#
Point in 2d#
\[
p = x\vec{u} + y\vec{v} + o
\]
\[
p = \begin{bmatrix}
x & y & 1
\end{bmatrix}\begin{bmatrix}
\vec{u} \\
\vec{u} \\
o
\end{bmatrix}
\]
Translation#
\[
p = \begin{bmatrix}
x & y & 1
\end{bmatrix}
\rightarrow
\begin{bmatrix}
x+a & y+b & 1
\end{bmatrix}
\]
\[
\begin{bmatrix}
x & y & 1
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
a & b & 1
\end{bmatrix} =
\begin{bmatrix}
x+a & y+b & 1
\end{bmatrix}
\]
Scaleing#
\[
\begin{bmatrix}
x & y & 1
\end{bmatrix}
\rightarrow
\begin{bmatrix}
ax & by & 1
\end{bmatrix}
\]
\[
\begin{bmatrix}
x & y & 1
\end{bmatrix}
\begin{bmatrix}
a & 0 & 0 \\
0 & b & 0 \\
0 & 0 & 1 \\
\end{bmatrix}=
\begin{bmatrix}
ax & by & 1
\end{bmatrix}
\]
TST#
- Translate to origin
- Scale
- Translate back
\[
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
-a & -b & 1
\end{bmatrix}
\begin{bmatrix}
Sx & 0 & 0 \\
0 & Sy & 0 \\
0 & 0 & 1 \\
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
-a & -b & 1
\end{bmatrix}
\]